JacobiElliptic is an implementation of Toshio Fukushima's & Billie C. Carlson's for calculating Elliptic Integrals and Jacobi Elliptic Functions .
The defaul algorithms are set to the Carlson algorithms.
Incomplete Elliptic Integrals
Function
Definition
F(φ, m)
$F(\phi|m)$ : Incomplete elliptic integral of the first kind
E(φ, m)
$E(\phi|m)$ : Incomplete elliptic integral of the second kind
Pi(n, φ, m)
$\Pi(n;\,\phi|\, m)$ : Incomplete elliptic integral of the third kind
J(n, φ, m)
$J (n;\, \phi |\,m)=\frac{\Pi(n;\,\phi|\, m) - F(\phi,m)}{n}$ : Associated incomplete elliptic integral of the third kind
Complete Elliptic Integrals
Function
Definition
K(m)
$K(m)$ : Complete elliptic integral of the first kind
E(m)
$E(m)$ : Complete elliptic integral of the second kind
Pi(n, m)
$\Pi(n|\, m)$ : Complete elliptic integral of the third kind
J(n, m)
$J (n|\,m)=\frac{\Pi(n|\, m) - K(m)}{n}$ : Associated incomplete elliptic integral of the third kind
Jacobi Elliptic Functions
Function
Definition
sn(u, m)
$\text{sn}(u | m) = \sin(\text{am}(u | m))$
cn(u, m)
$\text{cn}(u | m) = \cos(\text{am}(u | m))$
asn(u, m)
$\text{asn}(u | m) = \text{sn}^{-1}(u | m)$
acn(u, m)
$\text{acn}(u | m) = \text{cn}^{-1}(u | m)$